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Full text of "Men Of Mathematics"

THE MAN",   NOT THE METHOD
Hermite believed that numbers have an existence of their own
above all control by human beings* Mathematicians, he
thought, are permitted now and then to catch glimpses of the
superhuman harmonies regulating this ethereal realm of numer-
ical existence, just as the great geniuses of ethics and morals
have sometimes claimed to have visioned the celestial perfec-
tions of the Kingdom of Heaven.
It is probably right to say that no reputable mathematician
to-day who has paid any attention to what has been done in the
past fifty years (especially the last twenty-five) in attempting
to understand the nature of mathematics and the processes of
mathematical reasoning would agree with the mystical Her-
mite. Whether this modern scepticism regarding the other-
worldliness of mathematics is a gam or a loss over Hermite's
creed must be left to the taste of the reader. What is now almost
universally held by competent judges to be the wrong view of
'mathematical existence' was so admirably expressed by
Descartes in his theory of the eternal triangle that it may be
quoted here as an epitome of Hermite's mystical beliefs.
4I imagine a triangle, although perhaps such a figure does not
exist and never has existed anywhere in the world outside my
thought. Nevertheless this figure has a certain nature, or form,
or determinate essence which is immutable or eternal, which I
have not invented and which in no way depends on my mind.
This is evident from the fact that I can demonstrate various
properties of this triangle, for example that the sum of its three
interior angles is equal to two right angles, that the greatest
angle is opposite the greatest side, and so forth. Whether I
desire to or not, I recognize very clearly and convincingly that
these properties are in the triangle although I have never
thought about them before, and even if this is the first time I
have imagined a triangle. Nevertheless no one can say that I
have invented or imagined them.' Transposed to such simple
'eternal verities' as 1 -f- 2 = 3, 2 + 2 = 4, Descartes' everlast-
ing geometry becomes Hermite's superhuman arithmetic.
One arithmetical investigation of Hermite's, although rather
technical, may be mentioned here as an example of the pro-
phetic aspect of pure mathematics. Gauss, we recall, introduced
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